Finding Prime Fields of the Ring Z_3 x Z_5

I have a ring R = Z_3 x Z_5 and I have to find its two prime fields.

I have tried to find out what these minimal rings could be. However, if the ring must have elements (0,0) and (1,1), then it would also have to have elements a - b according to the subring criterion, which would cause the ring to have (0,0) - (1,1) = (-1,-1) = (2,4). Now also (2,4) - (1,1) would have to be in subring S, so eventually this would make the whole ring R. This leads to the fact that R has no subrings, which makes me confused, since the task is to find two of them.

If I make the assumption that the ring does NOT need to have (1,1) however, then I could get two subrings (which are also prime fields, I think?):
S1 ={ (0,0), (0,1), (0,2), (0,3), (0,4) } and
S2 = { (0,0), (1,0), (2,0) }

Can I assume that the subrings do not have (1,1)?
Are S1 and S2 prime fields of Z_3 x Z_5 ?

Could you please state your definition of "prime field"? I looked at a few sources and they all use "prime field" with respect to some field. Your ring is not a field, so it is not clear how to approach your problem.